Related papers: A Class of Orthohomological Triangles
In this paper we prove that if $P_1, P_2$ are isogonal points in the triangle $ABC$, and if $A_1B_1C_1$ and $A_2B_2C_2$ are their corresponding pedal triangles such that the triangles $ABC$ and $A_1B_1C_1$ are homological (the lines $AA_1,…
In 1840 Jacob Steiner on Christian Rudolf's request proved that a triangle with two equal bisectors is isosceles. But what about changing the bisectors to cevians? Cevian is any line segment in a triangle with one endpoint on a vertex of…
If P is a point inside triangle ABC, then the cevians through P extended to the circumcircle of triangle ABC create a figure containing a number of curvilinear triangles. Each curvilinear triangle is bounded by an arc of the circumcircle…
Given a triangle ABC, we derive the probability distribution function and the moments of the area of an inscribed triangle RST whose vertices are uniformly distributed on AB, BC, and CA. The theoretical results are confirmed by a Monte…
Given a right triangle ABC, with the ninety degree angle at A; consider the triangle O1OO2.Where the point O is the midpoint of the hypotenuseBC(and so the center of the triangle ABC's circumcircle), the point O1 being the triangle AOB's…
Two triangles are called orthologic if the perpendiculars from the vertices of one of them to the sides of the other are concurrent. In this paper, we explore the concept of orthology from various points of view. Mostly we work in terms of…
An N-tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC. In this paper we…
An N -tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile'". The tile may or may not be similar to ABC . This paper is the…
In this article we prove some theorems related to the triplets of triangles, homological two by two. These theorems will be used later to build triplets of triangles two by two tri-homological.
In this article we present a remarkable concyclicity of four centroids related to the orthocenters of the triangles $ABC,$ $BPC,$ $CPA,$ and $PAB$ of a quadrangle $ABCP$. Furthermore, we establish a result about orthologic triangles…
In this article we make the concept of a continuous family of triangles precise and prove the moduli functor classifying oriented triangles admits a fine moduli space but the functor classifying non-oriented triangles only admits a coarse…
This is an example on the cohomology of threefolds.
The edge-to-edge tilings of the sphere by congruent quadrilaterals of Type $a^2bc$ are classified as $3$ classes: a sequence of two-parameter families of $2$-layer earth map tilings with $2n$ $(n\ge3)$ tiles, a one-parameter family of…
In a previous paper we have introduced the ortho-homological triangles, which are triangles that are orthological and homological simultaneously. In this article we call attention to two remarkable ortho-homological triangles (the given…
Two triples of triangles having pairwise disjoint outlines in 3-space are called combinatorially isotopic if one triple can be obtained from the other by a continuous motion during which the outlines of the triangles remain pairwise…
Inscribed angles are investigated in taxicab geometry with application to the existence and uniqueness of inscribed and circumscribed taxicab circles of triangles.
An $N$-tiling of triangle $ABC$ by triangle $T$ (the `tile') is a way of writing $ABC$ as a union of $N$ copies of $T$ overlapping only at their boundaries. Let the tile $T$ have angles $(\alpha,\beta,\gamma)$, and sides $(a,b,c)$. This…
By counting with triangles and the octohedral axiom, we find a direct way to prove the formula of To\"en in \cite{Toen2005} for a triangulated category with (left) homological-finite condition.
An N -tiling of triangle ABC by triangle T is a way of writing ABC as a union of N triangles congruent to T, overlapping only at their boundaries. The triangle T is the "tile". The tile may or may not be similar to ABC . We wish to…
Right triangulated categories can be thought of as triangulated categories whose shift functor is not an equivalence. We give intrinsic characterisations of when such categories have a natural extriangulated structure and are appearing as…